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Research Papers

Adaptive Complex Function Projective Synchronization of Uncertain Complex Chaotic Systems

[+] Author and Article Information
Fangfang Zhang

School of Electrical Engineering
and Automation,
Qilu University of Technology,
No. 3501 Daxue Road,
Jinan 250353, China
e-mail: zhff4u@163.com

Shutang Liu

College of Control Science and Engineering,
Shandong University,
No. 17923 Jingshi Road,
Jinan 250061, China
e-mail: stliu@sdu.edu.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 1, 2014; final manuscript received June 18, 2015; published online August 12, 2015. Assoc. Editor: Mohammad Younis.

J. Comput. Nonlinear Dynam 11(1), 011013 (Aug 12, 2015) (11 pages) Paper No: CND-14-1277; doi: 10.1115/1.4030893 History: Received November 01, 2014

Complex function projective synchronization (CFPS) is the most general synchronization and it enhances the security of communication. However, there always exist unknown parameters for chaotic systems in the real world. Considering all possible cases of unknown parameters of two complex chaotic systems, we design adaptive CFPS schemes and parameters update laws based on speed-gradient (SG) method. The convergence factors and pseudogradient condition are added to regulate the convergence speed and increase robustness. SG method is extended from real field to complex field. Numerical simulations are performed to demonstrate the effectiveness and feasibility of the proposed schemes.

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Figures

Grahic Jump Location
Fig. 1

The CFPS process between drive system (44) and response system (46) with η1 = η2 = 0.2, η3 = 0.02, λ1 = 0.0001, λ2 = 0.001, λ3 = 0.00001, and γ1 = γ2 = γ3 = 10: (a) w1r(t),z1r(t); (b) w1i(t),z1i(t); (c) w2r(t),z2r(t); (d) w2i(t),z2i(t); and (e) w3(t),z3(t)

Grahic Jump Location
Fig. 2

The local enlarged CFPS process from 10 s to 20 s: (a) w1r(t),z1r(t); (b) w1i(t),z1i(t); (c) w2r(t),z2r(t); (d) w2i(t),z2i(t); and (e) w3(t),z3(t)

Grahic Jump Location
Fig. 3

The error dynamic of CFPS when B is known: (a) e1(t); (b) e2(t); (c) e3(t); (d) e4(t); and (e) e5(t)

Grahic Jump Location
Fig. 4

The identification process of A when B is known: (a) a1(t); (b) a2(t); and (c) a3(t)

Grahic Jump Location
Fig. 5

The CFPS process between uncertain systems (44) and (46) with η1 = η2 = 0.02, η3 = 0.01, λ1 = 0.0001, λ2 = 0.001, λ3 = 0.00001, τ1 = τ2 = τ3 = 0.001, τ4 = 0.0001, γ1 = γ2 = 0.0001, γ3 = 0.00001, γ4 = 0.000001, γ1 = γ2 = 1, and γ3 = 10: (a) w1r(t),z1r(t); (b) w1i(t),z1i(t); (c) w2r(t),z2r(t); (d) w2i(t),z2i(t); and (e) w3(t),z3(t)

Grahic Jump Location
Fig. 6

The error dynamic of CFPS between uncertain systems (44) and (46): (a) e1(t); (b) e2(t); (c) e3(t); (d) e4(t); and (e) e5(t)

Grahic Jump Location
Fig. 7

The identification process of A when B is unknown: (a) a1(t); (b) a2(t); and (c) a3(t)

Grahic Jump Location
Fig. 8

The identification process of B: (a) b1(t); (b) b2(t); (c) b3(t); and (d) b4(t)

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